Article pubs.acs.org/JPCB
Effect of Energy Polydispersity on the Nature of Lennard-Jones Liquids Trond S. Ingebrigtsen* and Hajime Tanaka* Institute of Industrial Science, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan ABSTRACT: In the companion paper [Ingebrigtsen, T. S.; Tanaka, H. J. Phys. Chem. B 2015, 119, 11052] the effect of size polydispersity on the nature of Lennard-Jones (LJ) liquids, which represent most molecular liquids without hydrogen bonds, was studied. More specifically, it was shown that even highly size polydisperse LJ liquids are Roskilde-simple (RS) liquids. RS liquids are liquids with strong correlation between constant volume equilibrium fluctuations of virial and potential energy and are simpler than other types of liquids. Moreover, it was shown that size polydisperse LJ liquids have isomorphs to a good approximation. Isomorphs are curves in the phase diagram of RS liquids along which structure, dynamics, and some thermodynamic quantities are invariant in dimensionless (reduced) units. In this paper, we study the effect of energy polydispersity on the nature of LJ liquids. We show that energy polydisperse LJ liquids are RS liquids. However, a tendency of particle segregation, which increases with the degree of polydispersity, leads to a loss of strong virial-potential energy correlation but is mitigated by increasing temperature and/or density. Isomorphs are a good approximation also for energy polydisperse LJ liquids, although particle-resolved quantities display a somewhat poorer scaling compared to the mean quantities along the isomorph. energy U equilibrium fluctuations at constant volume V. The virial-potential energy correlation is quantified via the correlation coefficient R given by
1. INTRODUCTION Fluids showing a distribution in a variable characterizing the constituent molecules are called polydisperse fluids.1,2 The distributed variable can be not only the size of the molecules but also their shape, charge, and more. A binary mixture composed of two different molecular sizes is a simple example of a polydisperse fluid. However, in general, the fluid can consist of numerous components. Bitumen is an example of a highly polydisperse fluid with millions of components and constitutes the “glue” part of asphalt owing to the unique and perplexing properties obtained from the highly polydisperse and viscous mixture.3 Polydisperse fluids are interesting from both a theoretical and an experimental perspective.4−25 In experiments, for instance, polydispersity can be used as a tool to avoid crystallization and can thus facilitate the study of highly viscous fluids.21,26 The reason for the latter is the higher nucleation barriers of the supercooled polydisperse fluid22 due to the increase of frustration against crystallization.27,28 Theoretically, polydisperse fluids pose a challenge as new perplexing phenomena, e.g., fractionation,8,9,12,14,15 can occur. Fractionation is the phenomenon that takes place when the fluid (or solid) forms additional phases, each showing a unique distribution. In fact, describing this phenomenon correctly has led to the creation of new theories as well as new simulation methods.9,12,14,15,29 In a recent paper30 we studied the effect of size polydispersity on the property of being a so-called Roskildesimple (RS) liquid.31−37 RS liquids are characterized by having strong correlations between virial W and potential © 2016 American Chemical Society
R=
⟨ΔW ΔU ⟩ ⟨(ΔW )2 ⟩ ⟨(ΔU )2 ⟩
(1)
in which Δ denotes deviation from mean value, ⟨...⟩ provides NVT ensemble averages, and −1 ≤ R ≤ 1. RS liquids are defined pragmatically32 by requiring R ≥ 0.90 and R depends on the state point. The class of RS liquids is believed to include most or all van der Waals and metallic liquids, but to exclude most or all covalent-bonding, hydrogen-bonding, strongly ionic, and dipolar liquids.32,37 RS liquids are simpler than other types of liquids displaying, e.g., isomorphs.35,38 Isomorphs are curves in the phase diagram of RS liquids along which structure, dynamics, and some thermodynamic quantities are invariant in dimensionless (reduced) units; reduced units refer in this case to macroscopic quantities, such as T and ρ−1/3, rather than the usual approach in simulations via microscopic quantities (see ref 35 for details). In the companion paper,30 we demonstrated that even highly size polydisperse Lennard-Jones (LJ) liquids are RS liquids. Furthermore, it was shown that these liquids have isomorphs to a good approximation. Real liquids, however, are Received: May 31, 2016 Revised: July 18, 2016 Published: July 19, 2016 7704
DOI: 10.1021/acs.jpcb.6b05486 J. Phys. Chem. B 2016, 120, 7704−7713
Article
The Journal of Physical Chemistry B dispersed in additional variables besides their size. As an example, colloidal fluids can have distribution in their size as well as their charge;39 polymer fluids need both a size and an energy distribution to be described correctly in simulations.40 In this paper we study energy polydisperse LJ liquids as a first step in understanding how these additional factors affect polydisperse RS liquids. In experiments, a polydispersity in the refractive index should provide a similar situation. An energy distribution has in previous studies been applied together with, e.g., a distribution in the molecular size12,40 and has only very recently been studied isolated for its interesting phase behavior.41−43 To fully unravel the relative effects of polydispersity on strong virial-potential energy correlation, we focus, in this study, on energy polydisperse LJ liquids.41 Although this system is more difficult to study in experiments than size polydisperse systems, we believe that the conclusions drawn from this kind of study are necessary for understanding the effect of polydispersity on the nature of LJ liquids. The paper is organized as follows. Simulation and model details are presented in section 2. Section 3 provides a brief introduction to RS liquids and their simple properties. Section 4 presents molecular dynamics (MD) computer simulation results. Finally, section 5 summarizes the paper and provides a brief outlook.
2. SIMULATION AND MODEL DETAILS MD computer simulations are utilized to investigate the effect of energy polydispersity on strong virial-potential energy correlation in the NVT ensemble.44 The RUMD package45 is used to simulate highly efficient GPU MD. A system size of N = 131 072 particles is used in the current study, and finite-size
Figure 2. Effect of polydispersity δ on the correlation coefficient R (eq 6) and density-scaling exponent γ ≡ ⟨ΔUΔW⟩/⟨(ΔU)2⟩ for the energy polydisperse LJ system (eq 2). The state points studied are described in the text. The text “Isomorph start” refers to section 4.2. (a) R as a function of δ. (b) γ as a function of δ.
effects were checked by simulating N = 62 500 and N = 5000 particles at selected state points. Finite-size effects were observed only for the N = 5000 system at very high polydispersities. The energy polydisperse LJ liquid has pair potential ⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤ vij(r ) = 4εij⎢⎜ ⎟ − ⎜ ⎟ ⎥ ⎝r⎠ ⎦ ⎣⎝ r ⎠
(2)
and total potential energy function U ≡ ∑i
